#147704

order	512
nilpotency class	4
isoclinism	isoclinic class of #137655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6)(7,17,20,33)(8,12)(9,15)(10,26)(11,16)(13,18,32,25)(14,21)(19,31,22,30)(23,28)(24,29)(27,35,36,34);(6,12)(7,18)(8,21)(9,10)(13,30)(14,15)(16,29)(17,27)(19,22)(20,25)(23,24)(26,28)(31,32)(33,36);(2,4)(5,7)(6,13)(8,22)(9,25)(10,18)(11,20)(12,27)(14,30)(15,33)(16,32)(17,26)(19,24)(21,35)(23,34)(28,31)(29,36)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239802, #72239807, #72239839, …
maximal normal subgroup of	#72239802, #72239807, #72239839, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54900);